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KAM, lower-dimensional, invariant tori, quasi-periodic motion, renormalization.
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\ifdim\lastskip0$, we will perform the following scaling of the non-toral part of the phase space
\bel{smu}
S_\mu(q,p,u,w) =(q,\mu p,\mu^{1/2}u,\mu^{1/2}w)\,.
\ee
Under such a transformation, a Hamiltonian $H$ is transformed as $H\mapsto
\frac{1}{\mu} H\circ S_\mu$. Notice that the Hamiltonian $H_0$ is invariant under this transformation.
Given a matrix $T\in\GL(d,\Rr)$, we will also perform the canonical
scaling
\bel{tscaling}
\TT(q,p,u,w) =(Tq,\bar T^{-1}p,u,w)\,,
\ee
where $\bar T=T^T$ is the transpose of $T$. This scaling is an essential part of the method and the choice of matrices $T$ has played an important role in the previous schemes. In~\cite{KK1}, we have constructed a renormalization scheme using a sequence of scaling matrices $T\in\SL(d,\Zz)$ generated by a multidimensional continued fraction algorithm by Khanin, Lopes Dias and Marklof~\cite{KLM}, based on an algorithm by Lagarias~\cite{Lagarias}. The results of~\cite{KLM} provide good bounds in the case of Diophantine vectors $\omega$, and allowed for the extension of the results of~\cite{Kocic} for two degree of freedom Hamiltonians to higher dimensions~\cite{KLM2}. In order to extend the results to Brjuno frequency vectors, we have developed a different method~\cite{KK2,Kocic2}, which uses non-integer matrices in $\GL(d,\Rr)$. In the present paper, we will perform this scaling with non-integer matrices in~$\SL(d,\Rr)$. On a technical level, the problem at hand is more complicated than the one considered in~\cite{KK2}, due to the existence of finitely many additional ``resonances''.
Our matrices $T$ will have the property that $T\omega=\eta^{-1}\omega$, where $00$ and $\CC>0$ such that
\be
|\omega\cdot\nu+\Omega\cdot V|>\CC |\nu|^{-\tau},
\ee
for all $\nu\in\Zz^d\backslash\{0\}$, and all $V\in\Zz^D$ with
$00$, of analytic vector fields (see Section~\ref{sec2}), close to the vector field $K$, defined by $K(q,p,u,w)=(\omega,0,-i\bar\Omega u,
i\bar\Omega w)$, where $\bar\Omega=\rm{diag}
[\Omega_1,\dots,\Omega_D]$.
We note that in paper we are interested in Hamiltonians that are real analytic in the original variables $(q,p,\xi,\eta)$, that is which in these variables satisfy $H\circ C^*=C^*\circ H$, where $C^*$ is the complex conjugation. In the new variables, this symmetry implies that the Hamiltonians have the following property
\be
H\circ C(q,p,u,w)=C^*\circ H(q,p,u,w)\,,
\ee
where $C(q,p,u,w)=(q^*,p^*,-iw^*,iu^*)$. It also translates to the following property of Hamiltonian vector fields $C^* Q X\circ C=X,$ where $Q$ is the linear transformation $Q(q,p,u,w)=(q,p,-iw,iu)$. We will call the vector fields real if they satisfy this property.
The main result of this paper is the following.
\begin{theorem}\label{main}
For every Brjuno vector $\omega\in\Rr^d$ and $\Omega\in\Rr^D$ Diophantine with respect to $\omega$ with different and nonzero components, there exists an open neighborhood $B$ of real analytic Hamiltonian vector fields $X\in\AA_\rho$ around $K$, and an analytic codimension $d+D$ manifold $\WW\subset B$, such that every Hamiltonian vector field $X\in\WW$ has an analytic invariant $d$-dimensional invariant torus with frequency vector $\omega$.
\end{theorem}
\begin{remark}
The manifold $\WW$ is the stable invariant manifold for the sequence of renormalization operators that we will construct. The unstable directions of the renormalization correspond to changes in frequency vectors $\omega$ and $\Omega$. One can use the above result to prove the existence of invariant $d$-dimensional tori in families of Hamiltonian vector fields intersecting the stable manifold. If one is interested in the invariant tori with frequency vectors parallel to $\omega$, the number of the necessary parameters can be reduced by $1$. As usual (see e.g.~\cite{KK1}), close to a non-degenerate Hamiltonian vector field, for which the $q$-component $X_q|_{p=0}$ is transversal to $\omega$, one can further reduce the number of parameters needed to prove the existence of invariant tori by $d-1$, by considering the translations in the $p$-variables.
\end{remark}
The paper is organized as follows. In Section~\ref{sec2}, we define the spaces of vector fields that we consider. Section~\ref{BSINGLE} contains the formulation and estimates on a single renormalization step. In Section~\ref{BCOMPOSE}, we construct a sequence of renormalization parameters and the stable manifold for the sequence of corresponding renormalization operators. Section~\ref{BTORI} contains the construction of lower-dimensional invariant tori for vector fields on the stable manifold of the renormalization transformations.
\section{Spaces of vector fields}\label{sec2}
Since in this paper we will perform the scaling of the torus $\Tt^d=(\Rr/2\pi\Zz)^d$ using non-integer matrices, it will be necessary to consider functions with periodicity of different lattices in $\Rr^d$. Let $\{e_1,\dots,e_d\}$ be a basis and $\ZZ=\{\sum_{i=1}^dz_ie_i|z_i\in\Zz\}$ be a lattice in $\Rr^d$. Let also $\VV$ be its dual
lattice, i.e.\ the set of points $v\in\Rr^d$ satisfying $\exp(iv\cdot
z)=1$ for all $z\in\ZZ$. If $\ZZ=(2\pi\Zz)^d$, then $\VV=\Zz^d$.
On $\complex^n$ we use norms $\|c\|=\sup_j|c_j|$ and
$|c|=\sum_j|c_j|$. For linear operators between normed linear
spaces, we will always use the operator norm, unless stated
otherwise. Denote by $D_\rho$, with $\rho>0$, the set of all points
$(x,y,u,w)$ in
$\complex^d\times\complex^{d}\times\complex^{D}\times\complex^{D}$
characterized by $\|\im x\|0$
such that
\bel{emptyRectangle}
|v_\spe|>L \quad {\rm or}\quad |v_\spa|\ge\ell\,,\qquad \forall
v\in\VV\setminus\{0\}\,.
\ee
In other words, all points in $\VV$, except for the origin, lie
outside the disk characterized by $|v_\spe|\le L$ and
$|v_\spa|0$ or zero otherwise; $y_i$-component is the
$(v,\kappa,\alpha,\beta)$ mode of this component of the vector
field; $u_j$-component is the
$(v,\kappa,\alpha,\beta-\delta_{j}^\ell e_\ell)$ mode of this
component of the vector field, whenever $\beta_j>0$ or zero
otherwise; $w_j$-component is the $(v,\kappa,\alpha-\delta_{j}^\ell
e_\ell,\beta)$ mode of this component of the vector field, whenever
$\alpha_j>0$ or zero otherwise. Here $i=1,\dots,d$, $j=1,\dots,D$;
$\delta_{i}^\ell$ is the Kronecker delta, i.e.\ $\delta_{i}^\ell=1$
if $i=\ell$ and zero otherwise;
$e_\ell=(\delta_{1}^\ell,\dots,\delta_{\dim}^\ell)$, where $\dim=d$ or
$\dim=D$, is the unit vector in the $\ell-$th coordinate direction of
the standard basis. We have used Einstein's notation, i.e.\ we assume
the summation over repeated up and down indices.
\end{definition}
\begin{remark}\label{BResonant}
Notice that the operators $-i\partial_x$ and $\SS_\mu^\ast$ commute
on the space of vector fields considered, and that modes of a vector
field are the joint eigenvectors of $-i\partial_x$ and $S$, where
$S$ is the generator of the one-parameter group of scalings
$\SS_\mu^\ast$. A mode of a vector field characterized by mode
indices $(v,\kappa,\alpha,\beta)$ is the joint eigenvector for
$(-i\partial_x,S)$ corresponding to the eigenvalues $(v,k)$, where
$k=\frac{|\alpha|+|\beta|}{2}+|\kappa|-1$. Sometimes, we will also
call $(v,k)$ with $v\in\VV$ and $k\in\{-1,-1/2,0,1/2,1,3/2,\dots\}$
the mode indices, refereing to all the mode indices
$(v,\kappa,\alpha,\beta)$, with
$\frac{|\alpha|+|\beta|}{2}+|\kappa|-1$ equal to $k$.
\end{remark}
\begin{remark}\label{BResonant}
Consider the Fourier-Taylor
expansion~\eqref{Fourier-Taylor-Hamiltonians} of a Hamiltonian. Each
mode of this Hamiltonian, i.e.\ each term in this expansion
characterized by a quadruplet
$(v,\kappa,\alpha,\beta)\in\VV\times\Nn_{0}^d\times \Nn_{0}^D\times
\Nn_{0}^D$, generates a single mode Hamiltonian vector field
characterized by $(v,\kappa,\alpha,\beta)$. The $(0,0,0,0)$ mode of any Hamiltonian vector field is zero.
\end{remark}
\begin{definition}
Let $I=\VV\times\Nn_{0}^d\times \Nn_{0}^D\times \Nn_{0}^D$. In the first renormalization step, let
$\iplus$ be the set of all $(v,\kappa,\alpha,\beta)\in I$ satisfying
$|\omega\cdot v|\le\sigma|v|$ when $v\ne 0$, or $\alpha=\beta$ when $v=0$, or
$k:=\frac{|\alpha|+|\beta|}{2}+|\kappa|-1>0$, and let $\iminus$ be
the complement of $\iplus$ in $I$. Define $\Iplus$ and $\Iminus$ to
be the projection operators onto the subsets of {\it resonant} and
{\it nonresonant} vector fields spanned by modes with mode indices
in $\iplus$ and $\iminus$, respectively. The {\it resonant} and {\it
nonresonant} parts of a vector field $X\in\AA_{\rho}$ are defined as
$\Iplus X$ and $\Iminus X$, respectively. In addition, we define
$\mean_{k}$, for $k\in\{-1,-1/2,0,1/2,1,3/2,\dots\}$, as the
projection operator onto the space spanned by modes with mode
indices $(0,\kappa,\alpha,\beta)$, with
$\frac{|\alpha|+|\beta|}{2}+|\kappa|-1$ equal to $k$. The torus
averaging operator is then defined by $\mean=\sum_{k}\mean_{k}\,$.
\end{definition}
\begin{lemma}\label{BContractionL} If $0L$ by condition \eqref{emptyRectangle}.
Consequently,
\bel{BAboundOne}
A\le-{\rho'}\Bigl(1-\frac{\rho''}{{\rho'}}\zeta\Bigr)|v_\spe|+k\ln(\mu)+(k+1)\ln\left(\frac{\chi^{-1}\rho''}{{\rho'}}\right)\le-{\rho'}(1-\zeta\eta)L-\ln(\mu)\,,
\ee
where we have again used that $\rho''\le\chi{\rho'}$, $\mu<1$ and $
-1\le k\le 0$. The second bound in \eqref{BContraction} now follows
by using \eqref{LdeltaCond}.
Next, consider the case $k>0$, i.e $k\ge 1/2$. Since
$\rho''(1/2)|\omega\cdot v|\mbox{ for all }V=\beta-\alpha\in\Zz^D \mbox{ with
}|V|\le 2\}$, and
$$
\gamma^{(1)}=\max_{(v,\kappa,\alpha,\beta)\in
J}\left\{1,\frac{\sigma}{|\omega\cdot v+\Omega\cdot
V|}\right\}\,,\qquad \gamma^{(2)}=\max_{(v,\kappa,\alpha,\beta)\in
J}\left\{2,\frac{\sigma|v|}{|\omega\cdot v+\Omega\cdot V|}\right\}
$$
\begin{proposition}\label{small_denominators}
For all $0\ne v\in\VV$ and $V\in\Zz^D$ with $|V|\le 2$, if
$|\omega\cdot v|>\sigma|v|$ and $|\omega\cdot v+\Omega\cdot
V|\ne 0$, then $|\omega\cdot v+\Omega\cdot
V|\ge\sigma/\gamma^{(1)}$ and $|\omega\cdot v+\Omega\cdot
V|\ge\sigma/\gamma^{(2)}|v|$.
\end{proposition}
\proof If $|\omega\cdot v|>\sigma|v|$ and $|\Omega\cdot
V| (1/2)|\omega\cdot v|$ and thus
$|\omega\cdot v+\Omega\cdot V|\ge (\sigma/2)| v|$. Using conditions
\eqref{emptyRectangle} and \eqref{LdeltaCond} and $L\ge 1$, we also
obtain $|\omega\cdot v|>\sigma$ and thus $|\omega\cdot v+\Omega\cdot
V|\ge \sigma/2$, in that case.
The number of modes with $|\Omega\cdot V|>(1/2)|\omega\cdot
v|>(1/2)\sigma| v|$ and $|V|\le 2$ is finite. So, if $|\omega\cdot
v+\Omega\cdot V|\ne 0$ then $|\omega\cdot v+\Omega\cdot
V|\ge\sigma/\gamma^{(1)}$ and $|\omega\cdot v+\Omega\cdot
V|\ge\sigma/\gamma^{(2)}| v|$.\qed
Let $\wh K Z=[K,Z]$.
Let $c_\Omega>0$ be the smaller of the $\min_i \{|\Omega_i|\}$ and $\min_{i\ne j} \{|\Omega_i-\Omega_j|\}$.
\begin{proposition}\label{BKXBound} If $\rho>0$, and $Z\in\Iminus\AA'_\rho$,
then
\bel{BKXBounds}
\|\Iminus[Z,K]\|_\rho\ge{\sigma\over\gamma}\|Z\|_\rho'\,,
\ee
where
$\gamma=\frac{2(\rho+1)}{\rho}\max\{\gamma^{(1)},\gamma^{(2)}\}$.
\end{proposition}
\proof Assume that $(v,\kappa,\alpha,\beta)$ belongs to $\iminus$. Then, $k\le 0$ and either $|\omega\cdot v|>\sigma|v|$ with $v\ne 0$ or $\alpha\ne\beta$ with $v=0$. Due to our choice of the
norm, it suffices to verify these bounds for a single mode
$Z_{v,\kappa,\alpha,\beta}$ of vector field $Z\in\Iminus\AA'_\rho$.
Notice that $\wh K Z_{v,\kappa,\alpha,\beta}=i(\omega\cdot\
v+\Omega\cdot(\beta-\alpha))Z_{v,\kappa,\alpha,\beta}$ and that $\wh
K$ commutes with $\Iminus$. If $\alpha\ne\beta$ with $v=0$, then $|\omega\cdot\
v+\Omega\cdot(\beta-\alpha)|>c_\Omega \ge\sigma/\gamma^{(1)}$. The previous bounds and
\clmp(small_denominators) show that if $Z\in\Iminus\AA'_\rho$ and
$Y=[Z,K]$, then $\|Z\|_\rho\le \gamma^{(1)}/\sigma \|Y\|_\rho$ and
\bel{BomegadotnuInv}
\sum_{j=1}^d\left\|{\partial_{x_j}}Z\right\|_\rho
\le{\gamma^{(2)}\over\sigma}\|Y\|_\rho\,,\qquad
\sum_{j=1}^{d}\left\|{\partial_{y_j}}Z\right\|_\rho
+\sum_{j=1}^{D}(\left\|{\partial_{u_j}}Z\right\|_\rho +\left\|{\partial_{w_j}}Z\right\|_\rho)\le{2\gamma^{(1)}\over
\rho\sigma}\|Y\|_\rho\,.
\ee
As a result we obtain \eqref{BKXBounds}. \qed
This proposition allows us to apply the normal form theorem of~\cite{KK1},
which directly implies the following lemma. The positive number
$\varrho<1$ that appears in the following is fixed throughout the
paper. It has the meaning of the step-dependent domain parameter $\rho$ of the initial
vector fields that we would like to renormalize.
Consider the equations
\bel{BElimEqu}
\Iminus(X+[Z,X])=0\,,\qquad \Iminus\UU_\ssX^\ast X=0\,.
\ee
Constants that we call universal, do not depend on any renormalization parameters.
\begin{lemma}\label{BElim} Let $\rho>0$, and
let $\rho'=\rho-(\sigma/\gamma)\varrho$. There exist universal
constants $C_1$ and $C_2$ such that for every vector field
$X\in\AA'_\rho\,$, if
\bel{BElimCond}
\|X-K\|'_\rho\le C_1(\sigma/\gamma)\,,\qquad \|\Iminus X\|_\rho\le
C_1(\sigma/\gamma)^2\,,\qquad
\ee
then there exists a vector field $Z\in\Iminus\AA'_\rho$ and a change
of coordinates $\UU_\ssX:D_{\rho'}\to D_\rho\,$, solving equation
\eqref{BElimEqu}, such that the vector field $\UU_\ssX^\ast X$
belongs to $\AA_{\rho'}\,$, and
\beal{BElimBounds}
\|Z\|'_\rho\,,\|\UU_\ssX-\id\|_{\rho'} &\le&
C_2(\gamma/\sigma)\|\Iminus X\|_\rho\,,\nonumber\\
\|\UU_\ssX^\ast X-X\|_{\rho'} &\le&
C_2(\rho-\rho')^{-1}(\gamma/\sigma)\|\Iminus X\|_\rho\,,\\
\|\UU_\ssX^\ast X-X-[Z,X]\|_{\rho'} &\le&
C_2(\rho-\rho')^{-3}(\gamma/\sigma)^3\|\Iminus
X\|_\rho^2\,.\nonumber
\eea
The map $X\mapsto\UU_\ssX$ is continuous in the region defined by
\eqref{BElimCond}, and analytic in its interior.
\end{lemma}
By construction, the map $X\mapsto\UU_\ssX^\ast X$ is type preserving, in particular Hamiltonian vector fields remain Hamiltonian under this transformation (see the discussion at the end of Section 5 of~\cite{KK1}).
Define the operator $\Pp$ as the projection operator
$\Pp=\Ee_{-1}+\Ee_{0}^{\alpha=\beta}$ onto the space spanned by resonant
vector fields that expand under scaling. Since in this paper we have restricted our consideration to Hamiltonian vector fields, $\mean_{-1}$ is simply the zero operator. Let the restriction of $\RR$ to this subspace be denoted by $\LL$. Here, $\Ee_0=\Ee_{0}^{\alpha=\beta}+\Ee_{0}^{\alpha\ne\beta}$ is the decomposition of $\Ee_0$ into projection operators onto the spaces characterized by $\alpha=\beta$ and $\alpha\ne\beta$, respectively. We will also define $\meanp=\mean\Iplus=\mean-\mean\Iminus$. Notice that the terms $\mean\Iminus X$ are eliminated in the first renormalization step once and for all, i.e.\ after one renormalization step, we have $\meanp=\mean$. In the following, $\AA_\rho$ denotes the subspace of Hamiltonian vector fields.
\begin{theorem}\label{BRRBoundsT} There exist universal constants $C,R>0$, such
that the following holds, under the given assumptions on
$L,\ell,\eta,\zeta,\gamma$ and $\mu$. Let $B$ be the open ball in
$\AA_\rho(\VV)$, with $(\sigma/\gamma)\varrho0$, such that the conditions \eqref{BElimCond}
in \clml(BElim) hold, whenever $X$ belongs to the domain $B$,
defined by $\|X-K\|_\varrho0$ and for all
$k=\frac{|\alpha|+|\beta|}{2}+|\kappa|-1\ge 1/2$. Summing over all
$k\ge 1/2$ to get a bound on
$\|(\meanp-\proj)\RR(X)\|_{\eta\rho'}\,$, and then adding
\eqref{BRRBOne}, yields
\bel{}
\|(\Id-\proj)\RR(X)\|_{\eta\rho'} \le
C_1\eta^{-2}\mu^{1/2}\bigl[\|(\Id-\proj) X\|_{\rho'}
+\|(\Id-\proj)(\UU_\ssX^\ast X-X)\|_{\rho'}\bigr]\,,
\ee
if $C_1$ is chosen sufficiently large. Using again the second bound
in \eqref{BElimBounds}, and the fact that $\Iminus\proj=0$ and
$\Iminus\meanp=0$, we obtain the second inequality in
\eqref{BRRBounds}.
By \clml(BContractionL), we also have a bound
\beal{BRRBThreeA}
\|\proj\RR(X)-\RR(\proj X)\|_{\eta\rho'}
&=&\eta^{-1}\|\TT^\ast\SS_\mu^\ast\proj(\UU_\ssX^\ast
X-X)\|_{\rho'}\nonumber\\ &\le& \eta^{-2}
\|\proj(\UU_\ssX^\ast X-X)\|_{\rho'}\,.
\eea
Using \clml(BElim), the norm on the right hand side of
\eqref{BRRBThreeA} can be estimated as follows:
\bel{BRRBThreeB}
\|\proj(\UU_\ssX^\ast X-X)\|_{\rho'} \le
C_2(\gamma/\sigma)^6\|(\Id-\meanp)X\|_\rho^2
+\|\proj[Z,X]\|_{\rho'}\,,
\ee
where $Z=\Iminus Z$ is the vector field described in
\eqref{BElimEqu}. We can check that for Hamiltonian vector fields $\proj[Z,\proj X]=0$,
and thus
\beal{BRRBThreeC}
\|\proj[Z,X]\|_{\rho'} &=&\|\proj[Z,(\Id-\proj)X]\|_{\rho'} \le
C_3(\gamma/\sigma)\|Z\|'_\rho\|(\Id-\proj)X\|_\rho \nonumber\\&\le&
C_4(\gamma/\sigma)^2 \|(\Id-\meanp)X\|_\rho\|(\Id-\proj)X\|_\rho\,.
\eea
Here, we have used the bound on $\|Z\|'_\rho$ from \clml(BElim).
Combining the last three equations yields the third inequality in
\eqref{BRRBounds}.
The bound concerning the restriction $\LL$ of $\RR$ to $\Pp\AA_\rho(\VV)$, is obvious if one notices that this restriction is linear operator $\LL=\eta^{-1}\TT^\ast$.
\qed
\section{Composed renormalization transformations}\label{BCOMPOSE}
We express the Brjuno condition on $\omega$ (and thus on $\VV$) in
terms of the summability of the series of numbers
\bel{BanDef}
a_n=\sum_{k=n}^\infty 2^{n-k}\Bigl[
2^{-k-\kappa}\ln(1/\Omega'_{k+\kappa})+(k+\kappa')^{-2}\Bigr]\,,\qquad
\Omega_n'=\min_{02$ are two
integer constants to be determined later.
It follows from the definition that $a_{n+1}/20$ and $2\sigma_nL_{n-1}\le\ell_{n-1}$, for all $n\in\Nn$, if $\kappa'>b^{-1}$.
\begin{proposition}\label{summability}
For any fixed $\kappa'>0$ and sufficiently large $\kappa$, one has
$\sum_{n=1}^\infty\sigma_n<1/2$.
\end{proposition}
\proof Notice that
$$\sigma_n0$ depending only on $\kappa'$. This makes the
sum $\sum_{n=1}^\infty\sigma_n$ finite, and by choosing
$\kappa$ sufficiently large, we can make this sum smaller than $1/2$.\qed
\begin{proposition}\label{Bindeed} If $v\in\VV_{n-1}$ is nonzero, then
either $|v_\spa|\ge\ell_{n-1}$ or $|v_\spe|> L_{n-1}\,$.
\end{proposition}
\proof Assume that $v\in\VV_{n-1}$ satisfies $0e^{-a_n2^{n+\kappa}}$, this yields
\bel{Bindeed}
|v_\spa|=\lambda_{n-1}^{-1}|\nu_\spa|
\ge\lambda_{n-1}^{-1}\Omega'_{n+\kappa}
>\lambda_{n-1}^{-1}e^{-a_n2^{n+\kappa}}=\ell_{n-1}\,,
\ee
as claimed. \qed
\begin{definition}
Let $\Omega_{n-1}:=\lambda_{n-1}^{-1} \Omega$, for $n\in\Nn$. Let
also
$$J_{n-1}^{^-}=\{(v,\kappa,\alpha,\beta)\in\iminus(\VV_{n-1})\,:\,
|\omega\cdot v|<2|\Omega_{n-1}\cdot V|\mbox{ for all
$V=\beta-\alpha$ with }|V|\le 2\},$$ and
$$
\gamma_{n}:=\frac{8}{\varrho\lambda_{n-1}}\max_{(v,\kappa,\alpha,\beta)\in
J_{n-1}^{^-}}\left\{2,\frac{\sigma_n\max\{1,|v|\}}{|\omega\cdot
v+\Omega_{n-1}\cdot V|}\right\}\,.
$$
\end{definition}
\begin{definition}\label{rhon}
The domain of analyticity of functions that are going to be
renormalized at the $n$-th step is determined by the parameter
\be
\rho_{n-1}=\varrho\lambda_{n-1}\left(1-\sum_{k=1}^{n-1}\frac{\sigma_k}{\lambda_{k-1}\gamma_k}\right).
\ee
\end{definition}
The numbers $\rho_{n}$ are positive due to \clmp(summability) and
the fact that $\gamma_k\lambda_{k-1}>1$. Moreover
$\rho_{n-1}>\varrho\lambda_{n-1}/2$.
\begin{proposition}\label{Gamman}
If $\Omega$ is Diophantine with respect to $\omega$, then there
exists a universal constant $\xi>0$, such that for all $n\in\Nn$,
\be
\gamma_n0$, if
$\kappa'$ and then $\kappa$ are chosen sufficiently large, then for
all $n\ge 1$,
\bel{BmuBound}
\mu_n\le Ce^{-N2^{n+\kappa}a_n}\,,\qquad \mu_n\le C2^{-Nn}\,,\qquad
\mu_n\le C\left(\frac{A_n}{A_1}\right)^N.
\ee
\end{proposition}
\proof Let $C>0$ and $N>0$ be arbitrary. Since
$a_{n+1}/2
a_n/A_1>(n+\kappa')^{-2}/A_1>e^{-c'n/A_1}>e^{-2^n/A_1}>e^{-2^{n+\kappa}/(NA_1)}$,
where the last inequality is valid for sufficiently large $\kappa$,
implies the third bound in \eqref{BmuBound}. \qed
\clmp(BmuBoundP) directly implies the following.
\begin{corollary}\label{corollary} Given any $C,N>0$, if
$\kappa'$ and then $\kappa$ are chosen sufficiently large, then for all
$n\ge 1$,
\bel{BmuBound}
\mu_n\le C\sigma_n^N\,,\qquad \mu_n\le C\eta_n^N\,,\qquad \mu_n\le
C\lambda_n^N\le C\eta_n^N\le C\zeta_n^N \,,\qquad\mu_n\le
C\gamma_n^{-N}\,.
\ee
\end{corollary}
\proof To obtain the first bound, we have also used the fact that for any $C,N>0$ we have $\mu_n0$, such that the $n$-th step renormalization operator
$\RR_n$ is a bounded analytic map from an open ball $B_{n-1}$ in
$\AA_{\rho_{n-1},n-1}$ of radius $r(\sigma_n/\gamma_n)^2$, centered
at $K$, into $\AA_{\rho_n,n}\,$, satisfying $\|\LL_n^{-1}\|\le 1$
and
\beal{BRRnBounds}
\|(\Id-\meanp)\RR_n(X)\|_{\rho_n} &\le&
C\eta_n^{-2}(\gamma_n/\sigma_n)^{2}\mu_n^{1/2}
\|(\Id-\meanp)X\|_{\rho_{n-1}}\,,\nonumber\\
\|(\Id-\proj)\RR_n(X)\|_{\rho_n} &\le&
C\eta_n^{-2}(\gamma_n/\sigma_n)^{2}\mu_n^{1/2} \|(\Id-\proj)X\|_{\rho_{n-1}}\,,\\
\|\proj\RR_n(X)-\RR_n(\proj X)\|_{\rho_n}
&\le&C\eta_n^{-2}(\gamma_n/\sigma_n)^{6}
\|(\Id-\meanp)X\|_{\rho_{n-1}}\|(\Id-\proj)X\|_{\rho_{n-1}}\,.\nonumber
\eea
\end{theorem}
\smallskip
In what follows, a domain $\DD_{n-1}$ for $\RR_n$ is a subset of the
ball $B_{n-1}$ described in \clmt(BRRnBoundsT), that is open in
$\AA_{\rho_{n-1},n-1}$ and contains the vector field $K$. Given a
domain $\DD_{n-1}$ for each $\RR_n\,$, the domain $\wt\DD_n$ of the
``composed'' renormalization operator
$\wt\RR_{n+1}=\RR_{n+1}\circ\wt\RR_{n}$, for $n\in\Nn$, with
$\wt\RR_1=\RR_1$, is defined recursively as the set of all vector
fields in the domain of $\wt\RR_n$ that are mapped under $\wt\RR_n$
into the domain $\DD_n$ of $\RR_{n+1}\,$. By \clmt(BRRnBoundsT),
these domains are open and non-empty, and the transformations
$\wt\RR_n$ are analytic on $\wt\DD_{n-1}$.
\begin{theorem}\label{BRRnComposeT} Let $00$ smaller than half the constant $r$ from
\clmt(BRRnBoundsT).
Define the transformations $R_n\,$ by
\bel{BRnDef}
R_n(Z)=r_n^{-1}\bigl[\RR_n(K+r_{n-1}Z)-K\bigr]\,,\qquad n\in\Nn\,.
\ee
Define the projection operators $\mean_n=\meanp$, $\proj_n=\proj$, for $n\in\Nn_0$.
The restriction $R_n\proj_{n-1}$ defines a linear map from
$\proj_{n-1}\AA_{\rho_{n-1},n-1}$ to $\proj_n\AA_{\rho_n,n}\,$, which will
be denoted by $L_n\,$. By \clmt(BRRnBoundsT), $R_n$ is analytic and
bounded on the ball $\|Z\|_\varrho<2$, and satisfies
\beal{BRBounds}
\|(\Id-\mean_n)R_n(Z)\|_{\rho_{n}}
&\le&\eps_n\|(\Id-\mean_{n-1})Z\|_{\rho_{n-1}}\,,\nonumber\\
\|(\Id-\proj_n)R_n(Z)\|_{\rho_{n}}
&\le&\vartheta_n\|(\Id-\proj_{n-1})Z\|_{\rho_{n-1}}\,,\\ \|\proj_n
R_n(Z)-R_n(\proj_{n-1} Z)\|_{\rho_{n}}
&\le&\varphi_n\|(\Id-\mean_{n-1})Z\|_{\rho_{n-1}}\|(\Id-\proj_{n-1})Z\|_{\rho_{n-1}}\,,\nonumber
\eea
where
\bel{BetpOne}
\eps_n=\vartheta_n=C\eta_n^{-2}(\gamma_n/\sigma_n)^{2}(\gamma_{n+1}/\sigma_{n+1})^2\mu_n^{1/2}\mbox{
and }
\varphi_n=C\eta_n^{-2}(\gamma_n/\sigma_n)^{6}(\gamma_{n+1}/\sigma_{n+1})^2\,.\nonumber
\ee
Here, $C\ge 1$ is a constant that may depend on $\varrho$, but not
on any other renormalization parameters. In addition, we have
$\|L_n^{-1}\|<1/4$. We will restrict $R_n$ to the domain
$D_{n-1}\subset\AA_{\rho_{n-1},n-1}\,$, defined by
\bel{BRnDomainDef}
\|\proj_{n-1} Z\|_{\rho_{n-1}}<1\,,\qquad
\|(\Id-\proj_{n-1})Z\|_{\rho_{n-1}}0$. Then we
will show that $\Gamma_0$ is an invariant torus for $X_0\,$.
For every $n\ge 0$, define $\BB_n$ to be the vector space
$\AA_0(\VV_n)$, equipped with the norm
\bel{BnNorm}
\|f\|'_n=r_n^{-1}\|f\|_{0,\VV_n}
=r_n^{-1}\sum_{v\in\VV_n}\|f_v\|\,,\qquad
r_n=\frac{\varrho}{4^n}\lambda_n\,.
\ee
Denote by $B_n$ the unit ball in $\id+\BB_n\,$, centered at the
identity function $\id$, and by $B_n/2$ the ball of radius $1/2$ in
the same space.
\begin{proposition}\label{BMMnContracts} If $\kappa'$ and
then $\kappa$ are chosen sufficiently large, then there exists an
open neighborhood $B$ of $K$ in $\AA_\varrho\,$, such that for every
$X\in\WW\cap B$, and for every $n\ge 1$, the map $\MM_n$ is well
defined and analytic, as a function from $B_n$ to $\BB_{n-1}\,$.
Furthermore, $\MM_n$ takes values in $B_{n-1}/2$, and
$\|D\MM_n(F)\|\le{1/3}\,$, for all $F\in B_n\,$.
\end{proposition}
\proof Clearly, $\MM_n$ is well-defined in some open neighborhood of
$\id$ in $\BB_n\,$, and
\bel{BMMnExpr}
\MM_n(F)=\id+g+(\UU_{n-1}-\id)\circ(\id+g)\,,\qquad
g=\tilde\TT_{n}\circ f\circ\tilde\TT_{n}^{-1}\,,
\ee
where $f=F-\id$. In order to estimate the norm of $\UU_{n-1}-\id$,
we can apply \clml(BElim), with $\rho'_{n-1}$ equal to
$\rho_{n-1}-(\sigma_{n-1}/\gamma_{n-1})\varrho\,$. By \clml(BElim)
and \clmt(BRRnComposeT), there exist a constant $C>0$, such that
\beal{BMMnCOne}
\|\UU_{n-1}-\id\|_{\rho_{n-1}'} &\le& C(\gamma_n/\sigma_n)\|\Iminus
X_{n-1}\|_{\rho_{n-1}} \le C(\gamma_n/\sigma_n)\psi_{n-1}^{1/2-\epsilon}
\|(\Id-\meanp)X\|_\varrho\nonumber\\
&\le&\psi_{n-1}^{1/2-2\epsilon}\|(\Id-\meanp)X\|_\varrho
\le\psi_n^{1/9}\,,
\eea
for some small $\epsilon>0$, for all $n>1$, and for all $X\in\WW\cap B$.
Here, we have used \clmp(BmuBoundP) and assumed that $\kappa'$ and
then $\kappa$ have been chosen sufficiently large, and that the
neighborhood $B$ of $K$ has been chosen sufficiently small
(depending on $\kappa'$ and $\kappa$). Though all steps in
\eqref{BMMnCOne} cannot be carried through if $n=1$, the final
estimate is also valid in that case.
The composition with $\id+g$ in Equation \eqref{BMMnExpr} is
controlled by \clmp(BTrivial), using that
$\|g\|_{0,\VV_{n-1}}\le\eta_n^{-1}r_n\|f\|'_n$ is less than
$\rho_{n-1}'/2$, since we assume that $F\in B_n$. By using that
$r_n/r_{n-1}=\eta_n/4\,$, we obtain
$\|g\|'_{n-1}\le\eta_n^{-1}\eta_n/4\le 1/4\,$. From \eqref{BMMnCOne}
we obtain $\|\UU_{n-1}-\id\|_{n-1}'\le r_{n-1}\psi_n^{1/16}\le 1/2$,
if $\kappa'$ and $\kappa$ have been chosen sufficiently large. These
estimates show that $\MM_{n-1}$ maps $B_n$ into $B_{n-1}/2$.
Now, we obtain a bound on the norm of the derivative map
\be
D\MM_n(F)\bar f=\bar g+D(\UU_{n-1}-I)\circ(I+g) \bar g,
\ee
where $\bar g=\tilde\TT_{n}\circ\bar f\circ\tilde\TT_{n}^{-1}$.
Since $\|g\|_{0,\VV_{n-1}}\le\rho_{n-1}/2$, and
\be
\|D(\UU_{n-1}-I)\|_{\rho_{n-1}'/2}\le
\frac{2}{\rho_{n-1}'}\|\UU_{n-1}-I\|_{\rho_{n-1}'}
\ee
we obtain a bound on this derivative norm analogous to
\eqref{BMMnCOne}. This, together with the fact that the inclusion
map from $B_n$ into $B_{n-1}$ is bounded in norm by $\eta_n/4$,
shows that $\|D\MM_n(F)\|\le{1/3}\,$, for all $n\ge 1$, and for all
$F\in B_n\,$. \qed
Below, we will make use of the following estimate on the difference
between the flow for $X$ and the flow for the constant vector field
$(\omega,0)$.
\begin{proposition}\label{BBasicFlowBoundP} {\rm\bf ~\cite{KK1}} Let $\tau$ be a positive real number
and $X$ a vector field in $\AA_\varrho\,$, such that
$\tau\|X-\omega\|_\varrho0$. As proved above, the right (and thus left) hand side
of this equation converges in $\AA_0$ to $\Gamma_0\,$. In addition,
$\Gamma_{0,k}\to\Gamma_0$ in $\AA_0\,$, and the convergence is
pointwise as well, by part $(i)$ of \clmp(BTrivial). Thus, since the
flow $\Phi_0^{t}$ is continuous, we have
$\Phi_0^{t}\circ\Gamma_0\circ\Phi_\omega^{-t}=\Gamma_0\,$. This
identity now extends to arbitrary $t\in\real$, due to the group
property of the flow, and the fact that composition with
$\Phi_\omega^s$ is an isometry on $\AA_0\,$.
Let $\AA_\rho^0$ be the subspace of functions $\AA_\rho$ which do
not depend on the variables $y,u,w$. In what follows, the torus
$\Gamma_0$ associated with a vector field $X\in\WW$ will be denoted
by $\Gamma_\ssX\,$. For convenience, we extend the map
$X\mapsto\Gamma_\ssX$ to an open neighborhood of $K$, by setting
$\Gamma_\ssX=\Gamma_\ssXp\,$, where $X'=(\Id+W)(X-\proj X)\in\WW$.
\begin{theorem}\label{BAnalyticTori} Let $\rho>\varrho+\delta$ with $\delta>0$.
Under the same assumptions as in \clmp(BMMnContracts), there exists
an open neighborhood $B$ of $K$ in $\AA_{\rho}(\VV_0)$, such that
$\Gamma_\ssX$ has an analytic continuation to $\|\im x\|